The non-existence of a regular maximal connected expansion of the reals

We prove that there is no regular maximal connected expansion of the Euclidean topology ε on the set R of real numbers. For this, we first consider a Hausdorff connected submaximal space (R,τ) with ε⊆τ and then, with the aid of the filter of the τ-dense sets, we define two specific expansions σ,τ⁎ of ε, such that τ⁎⊆σ,τ⁎⊆τ and (R,σ) is submaximal. We prove that if (R,τ) is in addition nearly maximal connected, then σ=τ⁎. Finally we prove that (R,τ) cannot be regular.